Following the introduction of a pathogen in a population of susceptible hosts, the nature of the "contact" between infected and non-infected members of the population becomes critical in shaping the outcome of the epidemics; nevertheless, the mechanisms leading to contact and transmission of common infectious diseases remain poorly understood. Here, we discuss how a combination of theoretical and experimental biofluid dynamic approaches can help shed light on the dynamics of contact for respiratory diseases.
In 1927 Kermack and McKendrick introduced and analyzed a rather general epidemic model (nota bene : their model takes the form of a nonlinear renewal equation and the familiar SIR model is but a very special case !). The aim of this lecture is to revive the spatial variant of this model, as studied in the late seventies by Horst Thieme and myself (see the AMS book 'Spatial Deterministic Epidemics' by L. Rass and J. Radcliffe, 2003).
The key result is a characterization of c_0 , the lowest possible speed of travelling waves and the proof that c_0 is also the asymptotic speed of epidemic propagation.
Mosquitoes can rapidly develop resistance to insecticides, which is a big problem in malaria control. Current insecticides kill rapidly on contact, but this leads to intense selection for resistance because young adults are killed. Of considerable current interest is the possibility of slowing down or even halting the evolution of resistance. Biologists believe that much weaker selection for resistance can be achieved if insecticides target only old mosquitoes that have already laid most of their eggs. This strategy aims to exploit the fact that most mosquitoes do not live long enough to transmit malaria, due to a long latency stage for the malaria parasite in the mosquito. I will present the results of some mathematical work using stage structured population models that can make predictions about the delayed onset of resistance in the mosquito population when they are subjected to an insecticide that only acts late in life. I will also summarise some ongoing work that includes the malaria disease dynamics and also the consequences of mosquito control using larvicides. Larvae can become resistant to larvicides, but the evolutionary cost of this acquired resistance may be reduced longevity as adults, which reduces the likelihood of the parasites completing their developmental stages and thus can actually benefit malaria control.
This is a joint collaboration with Rongsong Liu, Chuncheng Wang and Jianhong Wu.
Optimal control of two types of epidemic models with spatial and temporal features will be presented. Both examples will model rabies in raccoons. One model will be discrete in space and time and the other is a system of partial differntial equations.
The USA has suffered an epidemic of Lyme disease, which began in the late 1970s and continues to this day. In Canada, Lyme disease is an emerging infection due to recent expansion of the range of the tick vector Ixodes scapularis, which may in part be due to a warming climate. Here we describe how a comprehensive understanding of the ecology of I. scapularis, its hosts and the pathogens it transmits, have allowed us to predict the scope and direction of potential range expansion of Lyme disease with projected climate change. This knowledge has also allowed us to raise model-based hypotheses for how climate change may affect evolutionary processes of I. scapularis-borne microparasites and drive pathogen emergence. Together, these studies will allow us to limit the public health impact of Lyme disease and other zoonoses by prediction and early warning of tick-borne pathogen risk. However, we discuss here in general the strengths and limitations of modelling vector-borne disease spread for public health purposes.
In the last century, there has been an unprecedented increase in the numbers of emerging infectious diseases for humans. Approximately 60% of these diseases are naturally maintained by animals (i.e. zoonotic), primarily wildlife. Mathematical models are typically employed to predict how a disease will spread through a population. Yet, many of these models assume a homogenous population, in which all individuals are equally susceptible and have same probability of transmitting infection. Organisms exhibit heterogeneity in their ability to maintain pathogens; some individuals exhibit high pathogen loads (i.e. 'supershedders') whereas others maintain low pathogen loads. An individual's ability to maintain and amplify a pathogen may be determined by a suite of intrinsic and extrinsic factors, including age, sex, behavior, immunity, genetics, and environmental stressors. I will discuss our current research studying how variation in a bird's energetic condition, stress hormone profile, and migratory status affects their response to viral pathogens. Incorporating this heterogeneity into epidemic models will likely produce an outcome that is a better representation of reality.
Infectious disease ecology is perennially concerned with risks from new and emerging infectious diseases. SARS in humans, white nose syndrome in bats, and (possibly) colony collapse disorder in honey bees are just 3 examples from the last decade. But patterns of disease emergence do not appear to be spatially uniform. Rather, they seem to occur in locations where particular confluences of conditions favor establishment. In this talk, I will describe our investigations extending stuttering-chain emergence theories based on branching processes to include spatial heterogeneity and dispersal processes. The theory can be applied to a wide variety of dispersal models, and used to anticipate emergence hot-spots.
We develop spatial models of vector-borne disease dynamics on a network of patches to examine how the movement of humans in heterogeneous environments affects transmission. We show that the movement of humans between patches is sufficient to maintain disease persistence in patches with zero transmission. We construct two classes of models using different approaches: (i) Lagrangian that mimics human commuting behavior and (ii) Eulerian that mimics human migration. We determine the basic reproduction number R0 for both modeling approaches and study the transmission dynamics in terms of R0. We also study the dependence of R0 on some parameters such as the travel rate of the infectives.
Cholera was one of the most feared diseases of the 19th century, and remains a serious public health concern today. I will discuss some results on parameter identifiability and estimation for cholera models, present data from cholera epidemics in 19th century London, Angola 2006, and the current outbreak in Haiti, and discuss ongoing efforts to model these outbreaks.
There have been several recent outbreaks of cholera, which is a bacterial disease caused by the bacterium Vibrio cholerae. It can be transmitted to humans directly by person-to-person contact or indirectly via contaminated water. To better understand the dynamics of cholera, a general compartmental model is discussed that incorporates these two transmission pathways as well as multiple infection stages and pathogen states. A basic reproduction number is identified that gives a sharp threshold for the global dynamics and an estimate of the control required for eradication. Further models that incorporate temporary immunity and hyperinfectivity using distributed delays are formulated. Numerical simulations show that oscillatory solutions may occur for parameter values taken from the literature on cholera data.
The basic reproduction number and its computation formulae are established for epidemic models with reaction-diffusion structures. It is proved that the basic reproduction number provides the threshold value for disease invasion in the sense that the disease-free steady state is asymptotically stable if the basic reproduction number is less than unity and the disease is uniformly persistent if it is greater than unity. On the basis of these theoretical results, three epidemic models for rabies, lyme disease and West Nile transmissions are analyzed to reveal the better strategies for these diseases. With the aid of numerical simulations, we find that the reduction of heterogenous infection is beneficial because the more heterogenous infection leads to the higher value of basic reproduction numbers. Moreover, influences from spatial configurations of disease infection and diffusion coefficients are investigated. This is a joint work with Xiaoqiang Zhao.
Malaria is one of the most important parasitic infections in humans, and more than two billion people are at risk every year. To understand how the spatial heterogeneity and extrinsic incubation period of the parasite within the mosquito affect the dynamics of malaria epidemiology, we propose a nonlocal and time-delayed reaction-diffusion model. We then introduce the basic reproduction ratio for this model and show that it serves as a threshold parameter that predicts whether malaria will spread. A sufficient condition is obtained to guarantee that the disease will stabilize at a positive steady state eventually in the case where all the parameters are spatially independent. Further, we use two vaccination programs to simulate the efficiency of spatial control strategies. If time permits, I will also mention our more recent work on the global dynamics of an extended system, which incorporates a vector-bias term into this model. This talk is based on joint works with Drs. Yijun Lou and Zhiting Xu.
The high sickle cell gene frequency has been hypothesised to be related to the protective advantage against malaria disease among heterozygous individuals. The aim is to investigate the impact of malaria treatment on the frequency of sickle cell gene. For this, we develop a mathematical model that describes the interactions between malaria and sickle cell gene under malaria treatment. The model includes both homozygous for the normal gene (AA) and heterozygous for sickle cell gene (AS) and assumes that AS individuals are not treated since they do not show clinical symptoms. Our results indicate that the frequency of sickle cell gene decreases with an increase in the recovery rate of AA individuals. We conclude that malaria eradication strategies will lead to a reduction in sickle cell gene frequency since the gene will cease to provide protection to its carriers but become disadvantaged in the population.
Recent studies have suggested that the risk of exposure to Lyme disease is emerging in Canada because of the expanding range of I. scapularis ticks. The wide geographic breeding range of I. scapularis-carrying migratory birds is consistent with the widespread geographical occurrence of I. scapularis in Canada. However, how important migratory birds from the United States are for the establishment and the stable endemic transmission cycle of Lyme disease in Canada remains to be an issue of theoretical challenge and practical significance. In this paper, we design and analyze a periodic system of differential equations with a forcing term, to model the annual bird migration, to address the aforementioned issue. Our results show that ticks can establish in any migratory bird stopovers and breeding sites. Moreover, bird-transported ticks may increase the probability of B. burgdorferi establishment in an tick-endemic habitat.
Biological invasions are driven by the growth, reproduction, death, and dispersal of individual organisms. These processes are all stochastic and typically include nonlinear interactions. Simple models allow us to better understand invasions at the expense of some realism. Branching random walks provide a simple stochastic modeling framework that allows considerable insight into the interaction between individual-level processes, at the expense of not being able to incorporate interactions among individuals.
In previous work, we described the application of simple branching random walks for the study of biological invasions. Here we extend that analysis to multiple life-history stages. We show derivation of equations for the first and second moments, spatial extinction probabilities, and asymptotic invasion rates. The connections between the existing theory for deterministic biological invasions and branching random walks are highlighted. These ideas are illustrated with two examples drawn from the literature.
A multi-group cholera model is formulated that incorporates direct and indirect transmission with nonlinear incidence. This can be viewed as modeling spatial heterogeneity. The basic reproduction number $R_0$ is determined and shown to give a sharp threshold. In particular, if $R_0>1$, then the global stability of the endemic equilibrium is proved by using a graph-theoretic approach and a new combinatorial identity.
Discovered in Florida in 2005, Citrus Greening (Huanglongbing) has caused a major decline in citrus production in the state and now threatens Texas and California citrus growers as well. A brief history of this vector-transmitted disease in Florida will be provided along with the challenges facing the citrus industry and researchers. A grove-scale population model for citrus trees will be used to calculate the basic reproduction number under various assumptions. Mathematical results concerning the disease extinction and persistence will be presented.
The emergence of the highly pathogenic avian influenza (AI) sub-type H5N1 in many parts of the world has focused greater attention on the ecology of influenza in wild birds, as wild birds are the major natural reservoir for all known influenza A viruses. The persistence of avian influenza viruses in water, which is highly affected by environmental conditions, plays an important role in AI transmission in wild birds. We present mathematical models that include bird migration as well as the experimentally-derived relationships between viral persistence and environmental conditions. We demonstrate that the model is consistent with field survey data from Northern Europe. Our results show that environmental variations strongly affect both single season and long-term AI dynamics. In addition, environmental variations predict several interesting features of AI dynamics which are observed in real data: peak-time variation, place-to-place variation and seasonal double peaks (summer and fall). We further develop the model by incorporating spatial heterogeneity in environmental conditions, and find strong effects of environmental heterogeneity on AI dynamics.
A stage-structured periodic deterministic model was formulated to assess the climate warming impact on the tick ( Ixodes scapularis) population at Long Point, Ontario, Canada. The model was parametrized, and the tick development and questing activity data were complied from the laboratory and field experiments conducted in Canada. Mean monthly temperatures of the study region were estimated for the two periods between 1961-1990 and 2000-2009, and some Fourier series analysis was conducted to derive the season-based model coefficients by fitting the temperature and tick data. We analyze both analytically and numerically a stage-structured periodic deterministic model recently formulated to assess the climate warming impact on the tick Ixodes scapularis population invasion. We derive the basic reproduction number for the tick population, R_0, as the number of new female adult ticks produced by an index female adult tick when there are no density dependent constraints acting anywhere in the life cycle of the tick population. We confirm that the tick is doomed to extinction when R_0 < 1, and further find the successful tick invasion and the existence of a positive seasonal equilibrium state when R_0 > 1.
There are several strains of malaria protozoa spreading in different regions. On the other hand, this world becomes more highly connected by travels than ever before. This raises a natural concern of possible endemics of multiple strains in one region. This study, we try to use mathematical models explore such a possibility. Firstly, we propose a model to govern the within-host dynamics of multiple strains, and analysis of this model practically excludes the possibility of the co-infection (or super-infection) of a host by multiple strains. Then we move on to set up another model to describe the dynamics of the malaria between host and mosquito populations without supper-infection (using the result in Part I). By analyzing this model, we find that endemics of both strains in a single region is possible.